Research Article Modeling of Sand and Crude Oil Flow in ...downloads.hindawi.com/journals/je/2015/457860.pdf · Modeling of Sand and Crude Oil Flow in Horizontal Pipes during Crude

Research ArticleModeling of Sand and Crude Oil Flow in Horizontal Pipesduring Crude Oil Transportation

Samuel Eshorame Sanni1 A S Olawale2 and S S Adefila1

1Department of Chemical Engineering Covenant University Ota Nigeria2Department of Chemical Engineering Ahmadu Bello University Zaria Nigeria

Correspondence should be addressed to Samuel Eshorame Sanni adexz3000yahoocom

Received 31 October 2014 Accepted 4 December 2014

Academic Editor Tingyue Gu

Copyright copy 2015 Samuel Eshorame Sanni et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Some oil and gas reservoirs are often weakly consolidated making them liable to sand intrusion During upstream petroleumproduction operations crude oil and sand eroded from formation zones are often transported as a mixture through horizontalpipes up to the well heads and between well heads and flow stations The sand transported through the pipes poses seriousproblems ranging from blockage corrosion abrasion and reduction in pipe efficiency to loss of pipe integrity A mathematicaldescription of the transport process of crude oil and sand in a horizontal pipe is presented in this paper The model used to obtainthe mathematical description is the modified form of Doan et al (1996 and 2000) models Based on the necessity to introduce asand deposit concentration term in the mass conservation equation an additional equation for solid phase was derived Differenceformulae were generated having applied Fickrsquos equation for diffusion to the mass conservation equations since diffusion is one ofthe transport mechanisms Mass and volume flow rates of oil were estimatedThe newmodel when tested with field data gave 85accuracy at the pipe inlet and 97 accuracy at the exit of the pipe

1 Introduction

During upstream petroleum production practices rock oilsfrom reservoirs are often transported as a mixture withsand up to the well heads and from the well heads to flowstations At the head of the wells horizontal transmissionlines with or without screens transport the residual sand inthe oil from feeder lines to flow stations The entrained sandmay deposit on the walls of the pipe due to pressure dropcausing problems such as abrasion corrosion pipe blockagereduction in flow area loss in pipe integrity and mostimportantly low output from the lines [1] Sand exclusionmeasures (sand screens sand filters and gravel packs) usedhitherto are somewhat laborious and expensive [2] henceit is necessary to search for an alternative solution to theproblem such as using a mathematical model Popoola et al[3] discussed corrosion problems andmitigation of corrosionduring oil and gas production In this paper about eightcommonly encountered corrosion types as they relate tooil and gas production were mentioned alongside methods

of controlling them Amongst the methods suggested arematerials selection injection of inhibitors the applicationof protective coatings corrosion monitoring and inspectionand cathodic protection To date a model approach to sandcorrosion control is yet to be established however variousfluid-particle flowmodels were reviewed so as to make an aptchoice The paper of Srdjan et al [4] established an internalcorrosion prediction model for multiphase flow in a pipelinewhere a comprehensive CO

2H2S flow model was applied

in order to predict the effects of H2S water entrainment

corrosion inhibition and localized attack on a pipeline Themodel was validated using experimental data where effectof trace amount of H

2S on corrosion rate in the absence

of iron sulfide scales and the effects at the onset of ironsulfide scale formations were evaluated and measured at pHvalues less than 5 and equal to 6 at temperature of 20ndash80∘Cpressure of 1 to 77 bars and conditions of 119879 = 60∘C and77 bars respectively Van et al [5] gave a numerical sensitivityanalysis of the Wilson two-three-layer models for fully andpartially stratified flows They confirmed the validity of

Hindawi Publishing CorporationJournal of EngineeringVolume 2015 Article ID 457860 7 pageshttpdxdoiorg1011552015457860

2 Journal of Engineering

the two-layermodel for partially stratified flows but the three-layer model was found suitable for bed load motion wherefriction is significant Patankar and Joseph [6] in their workshowed the validation of a developed numerical scheme withexperiments using a bimodal suspension in a sedimentationcolumn The model was used to estimate sedimentationrates using two simulations with different grid sizes parcelnumber and time steps Frederic et al [7] modeled thesettling of solid particles embedded in a viscous fluid flowingunder gravity through a narrower section of a pipe Theystudied the effect of particle shape on relaxation time for bothdisk and rectangular shaped particles Glowinsky et al [8]model is useful for the direct numerical simulation of three-dimensional fluidization and sedimentation phenomenaThemodel suits well the Newtonian and non-Newtonian incom-pressible viscous flows past moving rigid bodies

Doan et al [1] model represents a simulation approach ofsand deposition inside a horizontal well Although the modelincludes channel height it can also account for the effect of oilviscosity and particle size on the transport process It is alsosuitable for calculating fluid and particles concentration andquantifying fluid delivery but cannot simulate the turbulenttransport of oil and sand Huang et al [9] focused on themotion of a two-dimensional circular cylinder inCouette andPoiseuille flows of a viscoelastic fluid Both neutrally buoyantparticles and nonneutrally buoyant particles were considered

Joseph [10] developed a general model for particulateflows The model incorporates only two types of forces in itsphase equations interaction and viscous forces The model issuitable for quantifying fluid delivery and can handle a widerange of particle loading and types Doan et al [11] model isa simulation of the movement of sand and crude oil insidea horizontal well Two fluids of different viscosities wereconsidered and the relationship between viscosity Reynoldsnumber drag coefficient and interaction coefficient wasdeterminedThemodel does not consider the effect of eddieswhich makes it unsuitable for turbulent transport of crudeoil and sand Therefore this paper seeks to cover the gapin knowledge by modifying the aforementioned Doan et almodels thus describing a new model for laminar and tur-bulent transport of sand and crude oil in a horizontal pipebetween the head of a well and its flow station

2 Model Modification

The Doan et al [1 11] models were developed for the case ofsand and oil flow in an oil well and this informed why theywere chosen from the reviewed models for application otherreasons being the inclusion of parameters such as sand andcrude oil concentration terms solid and liquid phase pres-sures solid and liquid densities liquid and solid interactionforces kinematic pressure liquid and solid concentrationssolid and liquid velocities and liquid viscosity among othersThemodel represented by (1)ndash(4) was subsequently modifiedby assuming that all other components (asphaltenes resinsand olefins) were dissolved in the oil at the flow conditionswhile taking effect of eddies into account

(flow of masses)

(force flows)

z z + dz

q120588(in) q120588(out)

Fin Fout

Figure 1 A typical pipeline system

21 The Doan et al Model Consider

120597

120597119905(120601) +

120597

120597119911(120601119908119904) = 0

(mass conservation equation for solid

phase-suspension)

(1)

120597

120597119905(120576) +

120597

120597119911(120576119908119891) = 0

(mass conservation equation for fluid

phase-suspension)

(2)

120597

120597119905(120601119908119904) +

120597

120597119911(120601119908119904119908119904)

= minus (120601119892) minus120601

120588119904

120597119875119904

120597119911+120573

120588119904

(119908119891minus 119908119904) minus119875119896

120588119904

120597120601

120597119911

(force equation for Solid phase)

(3)

120597

120597119905(120576119908119891) +

120597

120597119911(120576119908119891119908119891)

= minus (120576119892) minus120576

120588119891

120597119875119891

120597119911+120573

120588119891

(119908119891minus 119908119904)

(force equation for fluid phase)

(4)

22 Model Development and Modification ConsideringFigure 1 where a mixture of incompressible crude oil andsand flows through an element of length 119889119911 within a pipe oflength 119871 the conservation equations can be generated asfollows

mass or force flow into the systemminusmass or force flow outof the system = time rate of change of mass or momentum inthe system

This can be expressed mathematically as

119872in minus119872out =120597119872

120597119905 (5)

23 Model Assumptions Sand and oil do not mix Hencetheir mixing coefficients are ignored The particles are spher-ical and are of uniform size so that the same buoyancy effectwill be experienced by particles of the same size within aregion of flow andpipe sectionTheoil isNewtonianTheflowis isothermal Sand-oil suspension behaves as a continuumthat is sand particles behave like fluid and have a fluidizedvelocity hence sand molecules are not taken as a discreteentity but are in a continuous phase The fluid-particle

Journal of Engineering 3

interaction force and particle-particle interaction forces areof significance Sand particles movement is as a result ofthe surrounding liquid phase and pressure forces which areinherent as a result of inability of the liquid to stay whenit offers resistance to shear Gravity force is due to particlesweight in the carryingmedium Buoyancy force is interpretedas fluid-particle interaction force since liquid molecules arebeing displaced by descending solid particles acted upon bygravity and inertia force which keeps a body in its state of restormaintains itsmotionwhilemoving and kinematic pressurewhich is a particle-particle interaction force The deposit isconsidered entirely of sand phase that is other componentsof the oil that may add to the weight of sand that is resinsand olefins or those that have tendencies of being depositedsuch as asphaltenes are all considered soluble in the oilunder the flow conditions The pipe wall appeared somewhatsmooth hence surface roughness of the pipe was ignoredDiffusion is one of the major controlling mechanisms offluid-particle transport A coming paper will describe theeffect of mechanisms such as Euler and Froude numberson the flow behaviour

24 Application of Taylorrsquos Series Taylorrsquos series expansionformula was applied at the inlet and exit portions of thepipe to obtain the mass and force balance equations Thisincludes a third equation for solid phase and a sand depositconcentration term

Taylorrsquos series expansion formula is as follows

119891 (119911 + 119889119911) = 119891 (119911) +1198891199111198911015840(119911)

1+|119889119911|211989110158401015840(119911)

2 (6)

Truncating at the 2nd term gives

119891 (119911 + 119889119911) = 119891 (119911) +1198891199111198911015840(119911)

1 (7)

Since

119891 (119911) = 120590119902119904120588119904 (8)

then 1198891199111198911015840(119911)1 implies

1198891199111198911015840(119911) = 119889119911119891

1015840(120590119902119904120588119904) (9)

25 The New Model Consider

120597120601

120597119905+120597

120597119911(120601119908119904) = 0 (10)

120597120590

120597119905+120597

120597119911(120590119908119904) = 0 (11)

120597120576

120597119905+120597

120597119911(120576119908119891) = 0 (12)

120597

120597119905(Ψ119908119904) + (

120597

120597119911(Ψ119908119904119908119904))

= minus (Ψ119892) minusΨ

120588119904

120597119875119904

120597119911+120573

120588119904

(119908119891minus 119908119904) minus119875119896

120588119904

120597Ψ

120597119911

(13)

120597

120597119905(120576119908119891) + (

120597

120597119911(120576119908119891119908119891))

= minus (120576119892) minus120576

120588119891

120597119875119891

120597119911+120573

120588119891

(119908119891minus 119908119904)

(14)

The new model as compared to the Doan et al [1 11] modelsshows that (11) is an additional equation for solid phase while(13) is themodified form of (3) because it includes a total sandconcentration term that is Ψ

26 Model Calibration Correlations were used to obtainconstants such as molecular and eddy diffusivities Thecorrelations used include the following

(i) Correlation for Evaluation of Molecular Diffusivity Con-sider

119863 =1

6lowast 119889 lowast 119906

1015840 (15)

where 119863 = molecular diffusivity (coefficient of diffusion) 119889= diameter of particle 1199061015840 = average velocity of the entiremixture of sand and oil ((119908

119904+ 119908119891)2) and 119908

119904and 119908

119891=

nominal sand and fluid velocities respectivelyAs explained in Doan et al [1] fluid phase nominal

velocities of 50 cms and 68 cms correspond to an averageproduction rate of 110m3day and 150m3day respectivelyEquation (16) gives the flow rate of the mixture

119876 (Flow rate) = 119880 (velocity) lowast 119860 (cross-sectional area) (16)

which implies 119876 prop 119880

8025m3day 997888rarr 119908119898

(mix velocity) (17)

The sand and oil velocities were however scaled in orderto avoid numerical instability Now by calculation weobtain the nominal field velocity for a production rate of8025m3day

8025m3day 997888rarr 119908119898

150m3day 997888rarr 68ms

110m3day 997888rarr 50ms which implies

(8025 minus 150

150 minus 110) = (

119908119898minus 68

68 minus 50)

119908119898= 3004ms

(18)


Sand nominal velocity is assumed to be 90 of mix velocityso sand nominal velocity

(119908119904) = 09 lowast 3004ms = 2704ms

119889 = 005m

120601119908119904(sand velocity) = 006 lowast 2704ms = 16224ms

120576119908119891(fluid velocity) = 094 lowast 3004ms = 2834ms

119863 =1

6lowast 005 lowast

2834 + 16224

2= 01248m2s

(19)

But

119873119886 = 119863120597119862

120597119911

119863 prop1

119862(molecular diffusivity is inversely

proportional to change in concentration)

997904rArr 119863119862 = 119896

997904rArr 11986311198621= 11986321198622

(20)

where 1198631= diffusivity associated with 100 concentration

1198632= diffusivity associated with 6 concentration 119862

1= mix

concentration 1198622= sand concentration 119863

1= 02378m2s

1198621= 100119863

2= 119863119890 and 119862

2= 6

If1198632= 119863119890 then

119863119890(effective diffusivity) = 01248

006= 208m2s (21)

(ii) Correlation for Eddy Diffusivities Based on the workof Escobedo and Mansoori [12] within the limit of thesublaminar layer of fluid 119903+ le 5 and the eddy diffusivity wasevaluated using

1205761= (

(119903+)

1115)

3

lowast 120592 (22)

where 119903+ = dimensionless radial distance 119903 lt 119903+ lt 119903infin andby averaging we have (119903 rarr 119903

infin)2 = (0 + 5)2 = 25

The reason for averaging is because solid particles wereassumed to be of the same shape and size and for suchparticles in a region of flow it is easy to evaluate themeanmixvelocity and hence the mean dimensionless radial distance

But

120592 =120583

120588119891

(23)

where 120592 = kinematic viscosity 120583 = dynamic viscosity and 120588119891

= fluid density Consider

120583 = 00971 kgm sdot s

120588119891= 78443 kgm3

1205761= (

(25)

1115)

3

lowast00971

98443= 0000001395m2s

(24)

For the Buffer region

1205762= ((

(119903+)

114)

2

minus 01923) 120592 (25)

Here 5 le 119903+ le 30

119903+=5 + 30

2= 175 (calculated average value) (26)

The 1205763= (((175)114)

2minus 01923) lowast 0097278443 =

0000267893m2sIn the turbulent core region

1205763=(04119903+)

1lowast 120592 (27)

Here 119903+ ge 30Considering the whole range 119903+ le 30 and 119903+ ge 30 the

least value for 119903+ that can be obtained within the turbulentregion is 30 Hence an average value was arbitrarily obtainedIf the entire radial distance lies between 0 and 100 forparticles in a region of flow the average value for the turbulentcore should be 50 but other average values obtained havedecimal parts of 05 hence this value was reduced by 05

1205763= ((04 lowast 495)

1lowast00971

78443) = 0002450926m2s (28)

Now

120576119879= 1205761+ 1205762+ 1205763

= 0000001395 + 0000267893 + 0002450926

= 000272m2s

119863119879= 119863119890+ 120576119879= (208 + 000272) m2s = 208272m2s

(29)

Consider total diffusivity = sum of molecular and eddydiffusivities

27 Closure Problem Resolution(Ensuring Zero Degree of Freedom)

Note The new model would not have a solution because itconsists of five equations with eight unknown variables (120601120590 1206011015840 120576 119908

119904 119908119891119875119904 119875119891) However three constitutive equations

were introduced in order to resolve the closure problemTheyare

(i) 120590 + 120601 = Ψ

(ii) Ψ + 120576 = 1 and

(iii) 119875119904= 119875119894minus 119875

(30)

Considering Taylorrsquos series expansion form of the forceequation for solid phase and substituting 119875

119904= 119875119894minus 119875 results

are generated from simulation


Table 1 Values and variables used in the model

Parameter Field value Scaled valueSand and oil nominal velocities (2704 amp 3004) cms (2704 amp 3004) cmsChoke size 20 20Base sediment and water 1464 1464Tubing oil volume 80252m3d 80252m3dTubing head temperature 95∘C 95∘CTubing bottom temperature 8033∘C 8033∘CTubing head pressure 2457 bars 2457 barsProduced water flow rate 1826m3d 1826m3dSand diameter 150ndash200 microns 005mMass flow rate of sand 544E minus 05 gs 544 gsSand density 170544 kgm3 170544 kgm3

Oil viscosity 00971 kgmsdots 00971 kgmsdotsPipe diameter 544 inches (014m) 010m

Table 2 Similarities and differences between the new model and Doan et alrsquos model

Serial number Condition Doan et al models The modified Doan et al model

(1) Mass transport 1 solid phase equation + 1 liquidphase equation without eddies

2 solid phase equations + 1 liquid phaseequation which include eddy parameter

(2) Momentum transport Including a fluid parameter andsuspension parameter for sand

Including a fluid parameter andsuspension and deposition parameters forsand

(3) Mathematical solution Considering moleculardiffusivity term in its solution

Including both molecular and eddydiffusivity terms in the solution

(4) Number of equations (4) (5)

28 Finite Difference Method Difference formulae were gen-erated by first applying Fickrsquos equation for diffusion to themass conservation equations in order to proffer solution tothe model in terms of oil recovery

120601119894

119897+1= 120582 (120601

119894

119897+1minus 2120601119894

119897+ 120601119894minus1

119897) + 120601119894

119897(solid phase)

120576119894

119897+1= 120582 (120576

119894

119897+1minus 2120576119894

119897+ 120576119894minus1

119897) + 120576119894

119897(fluid phase)

(31)

where

1206011015840= 120601119908119904

1205761015840= 120576119908119891 120582 = minus

119863119879Δ119905

2Δ1199112

(32)

where 119863119879= total diffusivity Δ119905 = time change and Δ119911 =

change in axial distance

3 Model Validation

The simulation runs were carried out using data in Table 1within boundary conditions that is sand concentrations atthe inlet and outlet are 006 and 003 respectively while oilconcentrations are 094 and 097 at the pipe inlet and exitrespectively

gross oil in barrels per day = 7419 bbld

net oil in barrels per day = 6082 bbld

1 barrel of oil = 0158987m3d

After 24 hrs the simulation gave inlet concentration of094 and exit concentration of 0931495

Mass flow rate of oil = volume flow rate of oil lowast density ofoil

measured value minus calculated valuemeasured value

lowast 100 = error(33)

The data provided were well substituted into Taylorrsquos for-mulae obtained for the conservation equations in order todetermine the inlet and outlet mass and volume flow ratesalongside their corresponding errors At the inlet and outletthe model gave an accuracy of 85 and 97 respectively forcompared values (ie measured against the calculated value)of mass flow rates of oil while the compared outlet mass andvolume flow rates of oil yielded 97 accuracy each

4 Results

Table 2 shows similarities and contrasts between the Doanet al models and the new model


Table 3 Field values and calculated mass and volume flow rates of oil

Position Measured value Calculated value errorInlet mass flow rate of oil 1047 kgs 9144 kgs minus1486Inlet volume flow rate of oil 00117m3s 00137m3s minus308Outlet mass flow rate of oil 879 kgs 9061 kgs minus146Outlet volume flow rate of oil 00112m3s 001155m3s minus313

5 Discussion of Results

The new model includes a third equation and a depositionterm and incorporates the effect of eddies in its differenceformulae as contained in Table 2The additional equation thedeposition term and the incorporation of eddies are usefulfor estimating the sand deposit concentration within the pipeand to take care of the forces imposed on the particles by theconvectional currents of the turbulent stream At the end of24 hrs the oil influx at the pipe inlet in barrels per day was7419 bbld The equivalent mass flow rate of oil is 1047 kgsFrom the simulation the calculated mass flow rate of oil is9144 kgs yielding an error of 1486 when compared withthe measured value as shown in Table 3 The estimated errorconfirms that the modelrsquos accuracy in terms of quantifyingoil influx is about 85 Also at the inlet the measured andcalculated oil volume flow rates 00117m3s and 00137m3srespectively give a difference of 0002m3s whose errorestimate is ndash146This error estimate reveals that the modelis 85 accurate in terms of quantifying oil volume inflowConsidering the pipe exit the measured and calculated massflow rates of oil are 879 kgs and 906 kgs respectively whichgive a difference of 0371 kgs with a corresponding errorof ndash308 while the measured and calculated volume flowrates are 00112m3s and 001155m3s respectively givinga difference of 000035m3s with a corresponding error ofminus313 (see Table 3) The exit estimates for both cases (massand volume flow rate) prove the new model to be about 97accurate The difference in percent accuracies between theinlet and exit may be due to back-push or drawback on thestream at the elbow joint where the stream strikes the pipebefore it goes in Also crude oil contains gases (compressible)which may cause a change in density of the stream Since(mass)119898 = V lowast 120588 (product of volume and density) it impliesthat higher volume flows correspond to reduced densities andvice versa Furthermore themodel predictions from themassconservation equations reveal that the newmodel is valid andsuitable for turbulent transport operations of crude oil andsand in a horizontal pipe

6 Conclusions

The following conclusions were offered for this study

(i) A model has been developed that describes laminarand turbulent transport of sand and oil throughhorizontal pipes

(ii) The modelrsquos accuracy reveals that the new model canbe used to quantify oil recovery

(iii) The model can serve as an alternative sand manage-ment tool

Nomenclature

119860 Cross-sectional area (m2)119892 Gravitational acceleration (m sminus2)119875119891 Fluid phase pressure (kgmminus1 sminus2)

119875119896 Kinematic pressure (kgmminus1 sminus2)

119875119904 Solid phase pressure (kgmminus1 sminus2)

119902119891 Volume flow rate of oil (m3 sminus1)

119902119904 Volume flow rate of sand (m3 sminus1)119905 Time (hrs or s)120592119898 Volume of mix symbol adopted is ideal (m3)

119908119891 Oil velocity (m sminus1)

119908119904 Sand velocity (m sminus1)

119911 Axial distance (m)119871 Pipe length (m)120573 Fluid-particle interaction coefficient (kgm3 sminus1)Δ119911 Change in length (m)120576 Oil concentration (volume fraction) (ndash)120601 Suspended sand concentration (volume fraction) (ndash)120588119891 Oil density (kgm3)

120588119904 Sand density (kgm3)120590 Sand deposit concentration (ndash)119865in Force flow into system (kgms2)119865out Force flow out of system (kgms2)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by AFTRAC Oil Services inTrans Amadi Port Harcourt Nigeria

References

[1] Q Doan A Farouq A George andMOguztoreli ldquoSand depo-sition inside a horizontal wellmdasha simulation approachrdquo SPEJournal vol 39 pp 33ndash40 2000

[2] M Hackworth C Johnson J Heiland et al ldquoDevelopmentand first application of bistable expandable sand screenrdquo inProceedings of the SPE Annual Technical Conference and Exhibi-tion Proceedings-Mile HighMeeting of the Minds pp 1819ndash1832Denver Colo USA October 2003


[3] L T Popoola S A Grema G K Latinwo B Gutti and A SBalogun ldquoCorrosion problems during oil and gas productionand its mitigationrdquo International Journal of Industrial Chem-istry vol 4 article 35 2013

[4] N Srdjan C Jiyong and J Kun-Lin A Multiphase flow andInternal Corrosion Prediction Model For Mild Steel PipelinesInstitute for Corrosion and Multiphase TechnologyOhio Uni-versity Athens Ohio USA 2005

[5] E Van V Matous and S MiedemaTheoretical Description andNumerical Sensitivity Analysis on Wilson Model for HydraulicTransport of Solids in Pipelines Results from Research mimeo-graph Delft University of Technology Delft The Netherlands2001

[6] N Patankar and D Joseph ldquoSedimentation in a bi-modalsuspensionrdquo in Proceedings of the Annual World Congress pp1ndash16 AIChE New York NY USA 2001

[7] D Frederic L Moseni and M Hans ldquoParticle-in-cell solutionsfor creeping viscous flows with interfacesrdquo in Bifurcation andLocalization of Soils and Rocks 99 H Muhlhaus A Dyskin andE Pasternak Eds Balkema Rotterdam Netherlands 2001

[8] R Glowinsky T Pan T Hesla D Joseph and J PeriauxldquoDirect numerical simulationficticious domain approach forparticulate flowsrdquo Journal of Computational Physics vol 169 pp363ndash426 2001

[9] P Y Huang J Feng H H Hu and D D Joseph ldquoDirect simu-lation of the motion of solid particles in Couette and Poiseuilleflows of viscoelastic fluidsrdquo Journal of Fluid Mechanics vol 343pp 73ndash94 1997

[10] D Joseph ldquoFlow induced microstructures in newtonian andviscoelastic fluidsrdquo in Proceedings of the 5th World Congress ofChemical Engineering pp 1ndash16 AIChE San Diego Calif USAJuly 1996

[11] Q Doan A Farouq A George andM Oguztoreli ldquoSimulationof sand transport in a horizontal wellrdquo in Proceedings of the 2ndInternational Conference on Horizontal Well Technology pp 18ndash20 Calgary Canada November 1996

[12] J Escobedo and G A Mansoori ldquoSolid particle depositionduring turbulent flow production operationsrdquo in Proceedingsof the SPE Production Operation Symposium T X RichardsonEd Symposium Series no 29488 pp 439ndash446 Oklahoma CityOkla USA April 1995

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the two-layermodel for partially stratified flows but the three-layer model was found suitable for bed load motion wherefriction is significant Patankar and Joseph [6] in their workshowed the validation of a developed numerical scheme withexperiments using a bimodal suspension in a sedimentationcolumn The model was used to estimate sedimentationrates using two simulations with different grid sizes parcelnumber and time steps Frederic et al [7] modeled thesettling of solid particles embedded in a viscous fluid flowingunder gravity through a narrower section of a pipe Theystudied the effect of particle shape on relaxation time for bothdisk and rectangular shaped particles Glowinsky et al [8]model is useful for the direct numerical simulation of three-dimensional fluidization and sedimentation phenomenaThemodel suits well the Newtonian and non-Newtonian incom-pressible viscous flows past moving rigid bodies

Doan et al [1] model represents a simulation approach ofsand deposition inside a horizontal well Although the modelincludes channel height it can also account for the effect of oilviscosity and particle size on the transport process It is alsosuitable for calculating fluid and particles concentration andquantifying fluid delivery but cannot simulate the turbulenttransport of oil and sand Huang et al [9] focused on themotion of a two-dimensional circular cylinder inCouette andPoiseuille flows of a viscoelastic fluid Both neutrally buoyantparticles and nonneutrally buoyant particles were considered

Joseph [10] developed a general model for particulateflows The model incorporates only two types of forces in itsphase equations interaction and viscous forces The model issuitable for quantifying fluid delivery and can handle a widerange of particle loading and types Doan et al [11] model isa simulation of the movement of sand and crude oil insidea horizontal well Two fluids of different viscosities wereconsidered and the relationship between viscosity Reynoldsnumber drag coefficient and interaction coefficient wasdeterminedThemodel does not consider the effect of eddieswhich makes it unsuitable for turbulent transport of crudeoil and sand Therefore this paper seeks to cover the gapin knowledge by modifying the aforementioned Doan et almodels thus describing a new model for laminar and tur-bulent transport of sand and crude oil in a horizontal pipebetween the head of a well and its flow station

2 Model Modification

The Doan et al [1 11] models were developed for the case ofsand and oil flow in an oil well and this informed why theywere chosen from the reviewed models for application otherreasons being the inclusion of parameters such as sand andcrude oil concentration terms solid and liquid phase pres-sures solid and liquid densities liquid and solid interactionforces kinematic pressure liquid and solid concentrationssolid and liquid velocities and liquid viscosity among othersThemodel represented by (1)ndash(4) was subsequently modifiedby assuming that all other components (asphaltenes resinsand olefins) were dissolved in the oil at the flow conditionswhile taking effect of eddies into account

(flow of masses)

(force flows)

z z + dz

q120588(in) q120588(out)

Fin Fout

Figure 1 A typical pipeline system

21 The Doan et al Model Consider

120597

120597119905(120601) +

120597

120597119911(120601119908119904) = 0

(mass conservation equation for solid

phase-suspension)

(1)

120597

120597119905(120576) +

120597

120597119911(120576119908119891) = 0

(mass conservation equation for fluid

phase-suspension)

(2)

120597

120597119905(120601119908119904) +

120597

120597119911(120601119908119904119908119904)

= minus (120601119892) minus120601

120588119904

120597119875119904

120597119911+120573

120588119904

(119908119891minus 119908119904) minus119875119896

120588119904

120597120601

120597119911

(force equation for Solid phase)

(3)

120597

120597119905(120576119908119891) +

120597

120597119911(120576119908119891119908119891)

= minus (120576119892) minus120576

120588119891

120597119875119891

120597119911+120573

120588119891

(119908119891minus 119908119904)

(force equation for fluid phase)

(4)

22 Model Development and Modification ConsideringFigure 1 where a mixture of incompressible crude oil andsand flows through an element of length 119889119911 within a pipe oflength 119871 the conservation equations can be generated asfollows

mass or force flow into the systemminusmass or force flow outof the system = time rate of change of mass or momentum inthe system

This can be expressed mathematically as

119872in minus119872out =120597119872

120597119905 (5)

23 Model Assumptions Sand and oil do not mix Hencetheir mixing coefficients are ignored The particles are spher-ical and are of uniform size so that the same buoyancy effectwill be experienced by particles of the same size within aregion of flow andpipe sectionTheoil isNewtonianTheflowis isothermal Sand-oil suspension behaves as a continuumthat is sand particles behave like fluid and have a fluidizedvelocity hence sand molecules are not taken as a discreteentity but are in a continuous phase The fluid-particle





119891 (119911 + 119889119911) = 119891 (119911) +1198891199111198911015840(119911)

1+|119889119911|211989110158401015840(119911)

2 (6)


119891 (119911 + 119889119911) = 119891 (119911) +1198891199111198911015840(119911)

1 (7)

Since

119891 (119911) = 120590119902119904120588119904 (8)

then 1198891199111198911015840(119911)1 implies

1198891199111198911015840(119911) = 119889119911119891

1015840(120590119902119904120588119904) (9)


120597120601

120597119905+120597

120597119911(120601119908119904) = 0 (10)

120597120590

120597119905+120597

120597119911(120590119908119904) = 0 (11)

120597120576

120597119905+120597

120597119911(120576119908119891) = 0 (12)

120597

120597119905(Ψ119908119904) + (

120597

120597119911(Ψ119908119904119908119904))


120588119904

120597119875119904

120597119911+120573

120588119904

(119908119891minus 119908119904) minus119875119896

120588119904

120597Ψ

120597119911

(13)

120597

120597119905(120576119908119891) + (

120597

120597119911(120576119908119891119908119891))

= minus (120576119892) minus120576

120588119891

120597119875119891

120597119911+120573

120588119891

(119908119891minus 119908119904)

(14)




119863 =1


1015840 (15)


119904+ 119908119891)2) and 119908

119904and 119908

119891=





8025m3day 997888rarr 119908119898

(mix velocity) (17)


8025m3day 997888rarr 119908119898



(8025 minus 150

150 minus 110) = (

119908119898minus 68

68 minus 50)

119908119898= 3004ms

(18)



(119908119904) = 09 lowast 3004ms = 2704ms

119889 = 005m



119863 =1

6lowast 005 lowast

2834 + 16224

2= 01248m2s

(19)

But

119873119886 = 119863120597119862

120597119911

119863 prop1



997904rArr 119863119862 = 119896

997904rArr 11986311198621= 11986321198622

(20)



1= mix


1= 02378m2s

1198621= 100119863

2= 119863119890 and 119862

2= 6

If1198632= 119863119890 then


006= 208m2s (21)


1205761= (

(119903+)

1115)

3

lowast 120592 (22)


infin)2 = (0 + 5)2 = 25


But

120592 =120583

120588119891

(23)



120583 = 00971 kgm sdot s

120588119891= 78443 kgm3

1205761= (

(25)

1115)

3

lowast00971

98443= 0000001395m2s

(24)


1205762= ((

(119903+)

114)

2

minus 01923) 120592 (25)

Here 5 le 119903+ le 30

119903+=5 + 30


The 1205763= (((175)114)

2minus 01923) lowast 0097278443 =


1205763=(04119903+)

1lowast 120592 (27)



1205763= ((04 lowast 495)

1lowast00971

78443) = 0002450926m2s (28)

Now

120576119879= 1205761+ 1205762+ 1205763

= 0000001395 + 0000267893 + 0002450926

= 000272m2s

119863119879= 119863119890+ 120576119879= (208 + 000272) m2s = 208272m2s

(29)






(i) 120590 + 120601 = Ψ

(ii) Ψ + 120576 = 1 and

(iii) 119875119904= 119875119894minus 119875

(30)


119904= 119875119894minus 119875 results
















120601119894

119897+1= 120582 (120601

119894

119897+1minus 2120601119894

119897+ 120601119894minus1

119897) + 120601119894

119897(solid phase)

120576119894

119897+1= 120582 (120576

119894

119897+1minus 2120576119894

119897+ 120576119894minus1

119897) + 120576119894

119897(fluid phase)

(31)

where

1206011015840= 120601119908119904

1205761015840= 120576119908119891 120582 = minus

119863119879Δ119905

2Δ1199112

(32)



3 Model Validation










4 Results







6 Conclusions





Nomenclature












Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119891 (119911 + 119889119911) = 119891 (119911) +1198891199111198911015840(119911)

1+|119889119911|211989110158401015840(119911)

2 (6)


119891 (119911 + 119889119911) = 119891 (119911) +1198891199111198911015840(119911)

1 (7)

Since

119891 (119911) = 120590119902119904120588119904 (8)

then 1198891199111198911015840(119911)1 implies

1198891199111198911015840(119911) = 119889119911119891

1015840(120590119902119904120588119904) (9)


120597120601

120597119905+120597

120597119911(120601119908119904) = 0 (10)

120597120590

120597119905+120597

120597119911(120590119908119904) = 0 (11)

120597120576

120597119905+120597

120597119911(120576119908119891) = 0 (12)

120597

120597119905(Ψ119908119904) + (

120597

120597119911(Ψ119908119904119908119904))


120588119904

120597119875119904

120597119911+120573

120588119904

(119908119891minus 119908119904) minus119875119896

120588119904

120597Ψ

120597119911

(13)

120597

120597119905(120576119908119891) + (

120597

120597119911(120576119908119891119908119891))

= minus (120576119892) minus120576

120588119891

120597119875119891

120597119911+120573

120588119891

(119908119891minus 119908119904)

(14)




119863 =1


1015840 (15)


119904+ 119908119891)2) and 119908

119904and 119908

119891=





8025m3day 997888rarr 119908119898

(mix velocity) (17)


8025m3day 997888rarr 119908119898



(8025 minus 150

150 minus 110) = (

119908119898minus 68

68 minus 50)

119908119898= 3004ms

(18)



(119908119904) = 09 lowast 3004ms = 2704ms

119889 = 005m



119863 =1

6lowast 005 lowast

2834 + 16224

2= 01248m2s

(19)

But

119873119886 = 119863120597119862

120597119911

119863 prop1



997904rArr 119863119862 = 119896

997904rArr 11986311198621= 11986321198622

(20)



1= mix


1= 02378m2s

1198621= 100119863

2= 119863119890 and 119862

2= 6

If1198632= 119863119890 then


006= 208m2s (21)


1205761= (

(119903+)

1115)

3

lowast 120592 (22)


infin)2 = (0 + 5)2 = 25


But

120592 =120583

120588119891

(23)



120583 = 00971 kgm sdot s

120588119891= 78443 kgm3

1205761= (

(25)

1115)

3

lowast00971

98443= 0000001395m2s

(24)


1205762= ((

(119903+)

114)

2

minus 01923) 120592 (25)

Here 5 le 119903+ le 30

119903+=5 + 30


The 1205763= (((175)114)

2minus 01923) lowast 0097278443 =


1205763=(04119903+)

1lowast 120592 (27)



1205763= ((04 lowast 495)

1lowast00971

78443) = 0002450926m2s (28)

Now

120576119879= 1205761+ 1205762+ 1205763

= 0000001395 + 0000267893 + 0002450926

= 000272m2s

119863119879= 119863119890+ 120576119879= (208 + 000272) m2s = 208272m2s

(29)






(i) 120590 + 120601 = Ψ

(ii) Ψ + 120576 = 1 and

(iii) 119875119904= 119875119894minus 119875

(30)


119904= 119875119894minus 119875 results
















120601119894

119897+1= 120582 (120601

119894

119897+1minus 2120601119894

119897+ 120601119894minus1

119897) + 120601119894

119897(solid phase)

120576119894

119897+1= 120582 (120576

119894

119897+1minus 2120576119894

119897+ 120576119894minus1

119897) + 120576119894

119897(fluid phase)

(31)

where

1206011015840= 120601119908119904

1205761015840= 120576119908119891 120582 = minus

119863119879Δ119905

2Δ1199112

(32)



3 Model Validation










4 Results







6 Conclusions





Nomenclature












Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(119908119904) = 09 lowast 3004ms = 2704ms

119889 = 005m



119863 =1

6lowast 005 lowast

2834 + 16224

2= 01248m2s

(19)

But

119873119886 = 119863120597119862

120597119911

119863 prop1



997904rArr 119863119862 = 119896

997904rArr 11986311198621= 11986321198622

(20)



1= mix


1= 02378m2s

1198621= 100119863

2= 119863119890 and 119862

2= 6

If1198632= 119863119890 then


006= 208m2s (21)


1205761= (

(119903+)

1115)

3

lowast 120592 (22)


infin)2 = (0 + 5)2 = 25


But

120592 =120583

120588119891

(23)



120583 = 00971 kgm sdot s

120588119891= 78443 kgm3

1205761= (

(25)

1115)

3

lowast00971

98443= 0000001395m2s

(24)


1205762= ((

(119903+)

114)

2

minus 01923) 120592 (25)

Here 5 le 119903+ le 30

119903+=5 + 30


The 1205763= (((175)114)

2minus 01923) lowast 0097278443 =


1205763=(04119903+)

1lowast 120592 (27)



1205763= ((04 lowast 495)

1lowast00971

78443) = 0002450926m2s (28)

Now

120576119879= 1205761+ 1205762+ 1205763

= 0000001395 + 0000267893 + 0002450926

= 000272m2s

119863119879= 119863119890+ 120576119879= (208 + 000272) m2s = 208272m2s

(29)






(i) 120590 + 120601 = Ψ

(ii) Ψ + 120576 = 1 and

(iii) 119875119904= 119875119894minus 119875

(30)


119904= 119875119894minus 119875 results
















120601119894

119897+1= 120582 (120601

119894

119897+1minus 2120601119894

119897+ 120601119894minus1

119897) + 120601119894

119897(solid phase)

120576119894

119897+1= 120582 (120576

119894

119897+1minus 2120576119894

119897+ 120576119894minus1

119897) + 120576119894

119897(fluid phase)

(31)

where

1206011015840= 120601119908119904

1205761015840= 120576119908119891 120582 = minus

119863119879Δ119905

2Δ1199112

(32)



3 Model Validation










4 Results







6 Conclusions





Nomenclature












Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























120601119894

119897+1= 120582 (120601

119894

119897+1minus 2120601119894

119897+ 120601119894minus1

119897) + 120601119894

119897(solid phase)

120576119894

119897+1= 120582 (120576

119894

119897+1minus 2120576119894

119897+ 120576119894minus1

119897) + 120576119894

119897(fluid phase)

(31)

where

1206011015840= 120601119908119904

1205761015840= 120576119908119891 120582 = minus

119863119879Δ119905

2Δ1199112

(32)



3 Model Validation










4 Results







6 Conclusions





Nomenclature












Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














6 Conclusions





Nomenclature












Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Modeling of Sand and Crude Oil Flow in ...downloads.hindawi.com/journals/je/2015/457860.pdf · Modeling of Sand and Crude Oil Flow in Horizontal Pipes during Crude